Math 561: Theory of Probability I (Spring 2023)

This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics:

  1. basic limit theorems (law of large numbers, central limit theorem and large deviation principle);
  2. martingales and their applications;
  3. if time allows, we will give a brief introduction to Brownian motion and Stein's method for normal approximation.

Grades Canvas
Student hours 4-5:50pm Wednesdays + Appointment by email.
SyllabusClick here!

Instructor Partha Dey
Office 35 CAB
ContactBy email with subject line: "Math 561:"
Class TR 11:00am-12:20pm in room 147 Altgeld Hall.
Grader Andres Medina Landeros
Textbook I will post pdf lecture notes for each class.

Richard Durrett: Probability: Theory and Examples (Free Online edition v5). We will cover the first four chapters. It is okay to use another edition for studying. Some other relevent books:

P. Billingsley Probability and Measure (3rd Edition). Chapters 1-30 contain a more careful and detailed treatment of some of the topics of this semester, in particular the measure-theory background. Recommended for students who have not done measure theory.
Prerequisite The prerequisite for Math 561 is Math 540 - Real Analysis I. We will review measure theory topics as needed. Math 541 is nice to have, but not necessary.
DRESTo obtain disability-related academic adjustments and/or auxiliary aids, students should contact both the instructor and the Disability Resources and Educational Services (DRES) as soon as possible. You can contact DRES at 1207 S. Oak Street, Champaign, (217) 333-1970, or via e-mail at
Grading Policy Homework: 40% of the course grade. Homework will be assigned weekly on Thursdays on Canvas, to be submitted at the start of next Thursday lecture or earlier in Canvas. Solving a lot of problems is an extremely important part of learning probability. You are encouraged to work together on the homework, but I ask that you write up your own solutions and turn them in separately. Late homework will not be graded. If for some reason you've done a homework but can't turn it in online, send it via email before class. Because of this strict policy on late homework, I will drop your lowest score. Please talk to the instructor in cases of emergency.

Midterm: 20% will depend on an in-class midterm exam on Tuesday, March 28, 2023. It will be technically comprehensive, but emphasizing recent material up to the most recent graded and returned homework assignment. Exam problems will be similar to homework problems.

Final: 40% will depend on a take home final exam. The final take home exam will cover the most important topics of the whole course. It will be assigned on the last day of the class and will be due on (tentatively) Tuesday, May 9, 2023.

Week Date Due Content

1 Tu Jan 17 Probability Spaces.
Th Jan 19 Measures.

2 Tu Jan 24 Measures on Real Line.
Th Jan 26 HW 1 Random Variables and Distributions.

3 Tu Jan 31 Expectation.
Th Feb 2 HW 2 Properties of Expectation.

4 Tu Feb 7 Inequalities and Independence.
Th Feb 9 HW 3 Fubini's theorem & Pi-Lambda Theorem.

5 Tu Feb 14 Borel-Cantelli Lemmas.
Th Feb 16 HW 4 Strong Law of Large Numbers.

6 Tu Feb 21 Kolmogorov's maximal theorem.
Th Feb 23 HW 5 Applications of SLLN and 3-series theorem.

7 Tu Feb 28 Convergence in Distribution.
Th Mar 2 HW 6 Central Limit Theorems.

8 Tu Mar 7 Central Limit Theorems contd.
Th Mar 9 HW 7 Poisson Convergence.

9 Tu Mar 21 Helly's Selection Theorem.
Th Mar 23 HW 8 Characteristic functions.

10 Tu Mar 28 Midterm Sample midterm and solution, Midterm and Solution.
Th Mar 30 Conditional expectation.

11 Tu Apr 4 Stein's method Dependency Graph approach.
Th Apr 6 HW 9 Stein's method Exchangeble pair approach.

12 Tu Apr 11 Regular Conditional Probability.
Th Apr 13 HW 10 Martingales, Stopping time.

13 Tu Apr 18 Wald's Identities, Upcrossing inequalities.
Th Apr 20 HW 11 Martingale convergence, Maximal inequalities.

14 Tu Apr 25 Lp convergence for martingales.
Th Apr 27 HW 12 Reverse Martingales, Optional Stopping.

15 Tu May 2 Azuma-Hoeffding ineq., Concentration inequality.

16 T May 9 Final Take home Final (PDF, TeX)